Warm-UpLet f be a function such that lim f(2 + h) - f(2) = 5.

Which of the following must be true?hh 0

I. f is continuous at x = 2II. f is differentiable at x = 2III. The derivative of f is continuous at x = 2

(a) I only (b) II only(c) I and II only (d) I and III only(e) II and III only

Problem of the DayLet f be a differentiable function such that f(3) = 2 and f '(3) = 5. If the tangent line to the graph of f at x = 3 is used to find an approximation to a zero of f, that approximation is

A) 0.4B) 0.5C) 2.6D) 3.4E) 5.5

Problem of the DayLet f be a differentiable function such that f(3) = 2 and f '(3) = 5. If the tangent line to the graph of f at x = 3 is used to find an approximation to a zero of f, that approximation is

A) 0.4B) 0.5C) 2.6D) 3.4E) 5.5

Point (3, 2)Slope = 5Tangent y - 2 = 5(x - 3)Thus y = 5x - 13To find zero 0 = 5x - 13 x = 2.6

2-3: Product / Quotient Rules & Other Derivatives

©2002 Roy L. Gover (www.mrgover.com)

Objectives:•Learn and use the

product & quotient rules.•Derive derivatives of trignometric functions.•Use higher-order derivatives.

Important IdeaThe derivative of the product is not the product of the derivatives.

( ) ( ) ( ) ( )d d df x g x f x g x

dx dx dx

𝑇𝑟𝑦 𝑓 (𝑥 )=2𝑎𝑛𝑑𝑔 (𝑥 )=𝑥

If h(x) = f(x)g(x) what is the derivative?

If h(x) = f(x)g(x) what is the derivative?

f(x + Δx)g(x + Δx) - f(x)g(x)Δx

add a well chosen zero

f(x+Δx)g(x+Δx) + f(x+Δx)g(x) - f(x+Δx)g(x) - f(x)g(x)Δx

lim Δx 0

If h(x) = f(x)g(x) what is the derivative?

f(x + Δx)g(x + Δx) - f(x)g(x)Δx

add a well chosen zero

f(x+Δx)g(x+Δx) - f(x+Δx)g(x) + f(x+Δx)g(x) - f(x)g(x) Δx

f(x+Δx)(g(x+Δx) - g(x)) + g(x)(f(x+Δx) - f(x)) Δx

lim Δx 0

lim Δx 0

If h(x) = f(x)g(x) what is the derivative?

f(x + Δx)g(x + Δx) - f(x)g(x)Δx

add a well chosen zero

f(x+Δx)g(x+Δx) - f(x+Δx)g(x) + f(x+Δx)g(x) - f(x)g(x) Δx

f(x+Δx)(g(x+Δx) - g(x)) + g(x)(f(x+Δx) - f(x)) Δx

lim Δx 0

lim Δx 0

f(x+Δx)(g(x+Δx) - g(x)) Δx

lim Δx 0

+ g(x)(f(x+Δx) - f(x))Δx

lim Δx 0

If h(x) = f(x)g(x) what is the derivative?

f(x+Δx) (g(x+Δx) - g(x)) Δx

lim Δx 0

+ g(x) (f(x+Δx) - f(x))Δx

lim Δx 0

Evaluate limits

f(x) g'(x) + g(x) f '(x)

If h(x) = f(x)g(x) what is the derivative?

f(x) g'(x) + g(x) f '(x)

1st times derivative of second + 2nd times derivative

of 1st

Product Rule

(Rule extends to cover more than 2 factors)if j(x) = f(x)g(x)h(x) then

j'(x) = f '(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x)

The Product Rule

( ) ( )

( ) ( ) ( ) ( )

df x g x

dxd d

f x g x g x f xdx dx

Memori

ze

The Product RuleThe derivative of the product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.

ExampleFind the derivative, if it exists:

2 3( ) 2 1 1f x x x x

The product of two functions

ExampleFind the derivative, if it exists:

2 3( ) 2 1 1f x x x x

Important Idea

Be sure you simplify your answers by at least:

•combining like terms•eliminating negative exponents

Try ThisFind the derivative:

2( ) 3 2 5 4h x x x x 2'( ) 24 4 15h x x x

Can you use a method other than the product rule?

Example

Find the derivative:

Note the new notation for derivative

[ cos ]xD x x

In this example, you must use the product rule.

Try This

Find the derivative:[ sin ]xD x x

cos sinx x x

Important Idea

The derivative of a quotient is not the quotient of the derivatives.

The Quotient RuleThe derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator all divided by the denominator squared.

The Quotient Rule

2

( )

( )

( ) '( ) ( ) '( )

( )

d f x

dx g x

g x f x f x g x

g x

lo D hi minus hi D lo over lo2

We’ll prove this one next week…

ExampleFind the derivative using the quotient rule:

22( )

xf x

x

Is there an easier way…

ExampleFind the derivative using the quotient rule:

22( )

xf x

x

Is there an easier way…

Important Idea

Sometimes it is easier to re-write the function and find the derivative using rules other than the quotient rule.

Warm-UpFind the derivative using the quotient rule and simplify your answer:

3

2

3 2( )

1

x xf x

x

4 2

22

6 4 3'( )

1

x x xf x

x

Warm-UpFind the derivative using the quotient rule and simplify your answer:

3

2

3 2( )

1

x xf x

x

Example

2

5 2

1

d x

dx x

Must use the quotient rule on this one…

Warm-UpFind the derivative (hint: re-write and use the quotient rule):3 (1/ )

5

xy

x

Warm-UpFind the derivative (hint: re-write and use the quotient rule):3 (1/ )

5

xy

x

2

22

3 2 5

5

dy x x

dx x x

Try ThisFind the derivative (hint: re-write and use the quotient rule):

3 (1/ )

5

xy

x

Try ThisFind an equation of the line tangent to s(t) at t=2: 1

( )1

ts t

t

Example

tand

xdx

Find:

Try This

Find: secd

xdx

Hint: write and use the quotient rule.

1sec

cosx

x

sec tandy

x xdx

Do ThisMemorize the derivatives of

It’s on page 123…

Try ThisDifferentiate:

secy x x

' sec tan secy x x x x

What rule was used?

DefinitionIf you take the derivative of a derivative, you get a higher-order derivative. The notation is:

Second derivative:

''y"( )f x2

2

d y

dx2[ ]xD y

DefinitionIf you take the derivative of a derivative, you get a higher-order derivative. The notation is:

Third derivative:

'''y"'( )f x3

3

d y

dx

3[ ]xD y

Important IdeaThe first derivative represents a rate of change. The second derivative represents the rate of change of the rate of change. In physics, the first derivative is velocity; the second derivative is acceleration.

You need your book…

Again, b/c these are all over the AP

exam…

ExampleThe height above the ground of an object dropped from altitude is:

2( ) 16 2000s t t Find the velocity and acceleration of the object after 10 seconds.

ExampleThe height above the ground of an object dropped from altitude is:

2( ) 16 2000s t t Find the velocity and acceleration of the object after 10 seconds.

Lesson Close•Without using your notes, what is the product rule?

•Without using your notes, what is the quotient rule?

Assignment

126/1-43 odd,51,53,86